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Mirrors > Home > MPE Home > Th. List > zeneo | Structured version Visualization version GIF version |
Description: No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 11336 follows immediately from the fact that a contradiction implies anything, see pm2.21i 115. (Contributed by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
zeneo | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4587 | . . . . 5 ⊢ (𝐴 = 𝐵 → (2 ∥ 𝐴 ↔ 2 ∥ 𝐵)) | |
2 | 1 | anbi1d 737 | . . . 4 ⊢ (𝐴 = 𝐵 → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) ↔ (2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵))) |
3 | pm3.24 922 | . . . . 5 ⊢ ¬ (2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵) | |
4 | 3 | pm2.21i 115 | . . . 4 ⊢ ((2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵) |
5 | 2, 4 | syl6bi 242 | . . 3 ⊢ (𝐴 = 𝐵 → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)) |
6 | 5 | a1d 25 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))) |
7 | neqne 2790 | . . 3 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
8 | 7 | 2a1d 26 | . 2 ⊢ (¬ 𝐴 = 𝐵 → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))) |
9 | 6, 8 | pm2.61i 175 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 2c2 10947 ℤcz 11254 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 |
This theorem is referenced by: (None) |
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